REGULARIZATION OF THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION ON A TWO-DIMENSIONAL BOUNDED DOMAIN

In this paper, the problem of continuation of the ill-posed Cauchy problem solution’s is studied for matrix factorizations of the Helmholtz equation in a two-dimensional bounded domains. It is assumed that the solution of the problem exists and also it is continuously differentiable in a closed domain with exactly given Cauchy data. In this case, an explicit formula for the continuation of the solution as well as a regularization formula are established, under the indicated conditions. Their continuous approximations with a given error in the uniform metric are given instead of the Cauchy data. Additionally a stability estimate is obtained for the solution of the Cauchy problem in the classical sense.